Gruber P. Convex and Discrete Geometry

Given C and L, the problem arises to determine the numbers
23 Successive Minima 381
ϱ(C, L) = max � ϱ>0 :{ϱC + l : l ∈ L} is a lattice packing � ,
µ(C, L) = min � µ>0 :{µC + l : l ∈ L} is a lattice covering � ,
the packing radius and the covering radius of C with respect to L. Fromamore
arithmetic viewpoint, 2ϱ(C, L) = λ1(C, L) and µ(C, L) are called the homogeneous
and the inhomogeneous minimum of C with respect to L.
There are several inequalities between the quantities ϱ(C, L), µ(C, L), λ1(C, L),
...,λd(C, L), respectively, called transference theorems since they transfer information
from one situation to another situation, for example from packing to covering
and vice versa. A first result of this nature is the following transference theorem
of Jarník [543]. For a different transference theorem due to Kneser [601], see
Theorem 26.2.
A basic problem in Diophantine approximation is the simultaneous approximation
of real numbers by rationals. Equally important is the related problem of approximation
of linear forms. Compare Corollaries 22.3 and 22.4 to get an idea of such
results.
In the following we present Jarník’s transference theorem and a deep result of
Perron and Khintchine, which relates the approximation of d real numbers by rationals
with common denominator and the approximation of the linear form with these
reals as coefficients.
Jarník’s Transference Theorem
Our aim is to show the following estimates:
Theorem 23.4. Let C be an o-symmetric convex body and L a lattice in E d . Further,
let λi = λi(C, L) for i = 1,...,d and µ = µ(C, L). Then
1
2 λd ≤ µ ≤ 1
2 (λ1 +···+λd).
Proof. Left-hand inequality: Let {b1,...,bd} be a basis of L. The definition of µ
shows that there are vectors li ∈ L such that 1 2 bi − li ∈ µ C or bi − 2li ∈ 2µC
for i = 1,...,d. The vectors b1 − 2l1,...,bd − 2ld are linearly independent. (For
otherwise, there is a linear combination of these vectors with integer coefficients not
all 0 which is equal to o. Hence there are integers u1,...,ud, with greatest common
divisor 1, such that we have the equality � uibi = 2 � uili. Since b1,...,bd form
a basis, the point � uibi is primitive, and this equality cannot hold.) The definition
of λd then shows that λd ≤ 2µ, concluding the proof of the left-hand inequality.
Right-hand inequality: It is sufficient to show that
�
1
(1)
2 (λ1
�
+···+λd)C + l : l ∈ L is a lattice covering.